with psuedo-metric \(\rho (\mathbf {x},\mathbf {x}^{\prime })=\sqrt{{\rm Var}(Z(\mathbf {x})-Z(\mathbf {x}^{\prime }))}\)  for \(\mathbf {x},\mathbf {x}^{\prime }\in \mathcal {X}\) . By Borell-TIS inequality (cf. Proposition A.2.1 in [1]}), we have for any \(\epsilon >0\) ,
\(\Pi _n\left(\left| \Vert f-\hat{f}_n\Vert _\infty -\mathbb {E}\Vert f-\hat{f}_n\Vert _\infty \right|>\epsilon \,\big |\,\mathbb {D}_n\right)\le 2\exp \left(-\epsilon ^2/2\Vert \hat{V}_n\Vert _\infty \right).\) 
